Questions: A new car is purchased for 20800 dollars. The value of the car depreciates at 10.75% per year. What will the value of the car be, to the nearest cent, after 13 years?

A new car is purchased for 20800 dollars. The value of the car depreciates at 10.75% per year. What will the value of the car be, to the nearest cent, after 13 years?

Transcript text: A new car is purchased for 20800 dollars. The value of the car depreciates at $10.75 \%$ per year. What will the value of the car be, to the nearest cent, after 13 years?

Solution

Solution Steps

Step 1: Initial Value and Depreciation Rate

Let the initial value of the car be $ V_0 = 20800 $ dollars. The annual depreciation rate is given as $ r = 10.75\% = \frac{10.75}{100} = 0.1075 $.

Step 2: Number of Years

The number of years for which the car depreciates is $ t = 13 $.

Step 3: Future Value Calculation

To find the future value $ V_t $ of the car after $ t $ years, we use the formula for exponential decay: \[ V_t = V_0 \times (1 - r)^t \] Substituting the known values: \[ V_t = 20800 \times (1 - 0.1075)^{13} \]

Step 4: Simplifying the Expression

Calculating $ (1 - 0.1075) $: \[ 1 - 0.1075 = 0.8925 \] Thus, the expression becomes: \[ V_t = 20800 \times (0.8925)^{13} \]

Step 5: Final Calculation

Calculating $ (0.8925)^{13} $ and then multiplying by $ 20800 $: \[ V_t \approx 20800 \times 0.2271 \quad (\text{approximate value of } (0.8925)^{13}) \] This results in: \[ V_t \approx 4742.10 \]

Step 6: Conclusion

The value of the car after 13 years is approximately $ 4742.10 $ dollars.